text: Box models are widely used to model biogeochemical systems. Box models are simplified versions of complex systems, reducing them to boxes (or storage reservoirs) for chemical materials, linked by material fluxes (flows). Simple box models have a small number of boxes with properties, such as volume, that do not change with time. The boxes are assumed to behave as if they were mixed homogeneously. These models are often used to derive analytical formulas describing the dynamics and steady-state abundance of  the chemical species involved.
  The diagram at the right shows a basic one-box model. The reservoir contains the amount of material M under consideration, as defined by chemical, physical or biological properties. The source Q is the flux of material into the reservoir, and the sink S is the flux of material out of the reservoir. The budget is the check and balance of the sources and sinks affecting material turnover in a reservoir. The reservoir is in a steady state if Q = S, that is, if the sources balance the sinks and there is no change over time.
  The turnover time (also called the renewal time or exit age) is the average time material spends resident in the reservoir. If the reservoir is in a steady state, this is the same as the time it takes to fill or drain the reservoir. Thus, if τ is the turnover time, then τ = M/S. The equation describing the rate of change of content in a reservoir is
  When two or more reservoirs are connected, the material can be regarded as cycling between the reservoirs, and there can be predictable patterns to the cyclic flow. More complex multibox models are usually solved using numerical techniques.
  The diagram above shows a simplified budget of ocean carbon flows. It is composed of three simple interconnected box models, one for the euphotic zone, one for the ocean interior or dark ocean, and one for ocean sediments. In the euphotic zone, net phytoplankton production is about 50 Pg C each year. About 10 Pg is exported to the ocean interior while the other 40 Pg is respired. Organic carbon degradation occurs as particles (marine snow) settle through the ocean interior. Only 2 Pg eventually arrives at the seafloor, while the other 8 Pg is respired in the dark ocean. In sediments, the time scale available for degradation increases by orders of magnitude with the result that 90% of the organic carbon delivered is degraded and only 0.2 Pg C yr is eventually buried and transferred from the biosphere to the geosphere.